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112       XIII   INTEGRATION

9.   If/is any real-valued function on 1= [0, 1], the nth Bernstein polynomial offis
defined to be

(a)    If /^ 0 on I, then Bn / i> 0 on I, and Bn i is equal to the constant 1 . Deduce that
HB.,,11 ^ H/ll in *<I).

(b)    Show by induction on k that, if fk(t) = /*, we have

(**)                                      Bn,/fc(0 = <Wfc + P*,n(0,

where akr n = (1 — (k — l)/ri)ak-ltn and <30>n = 1, and P*,n is a polynomial of degree
<^A; — 1 whose coefficients are less than Ck/n in absolute value, where C* is a constant
independent of n. (Differentiate the equation (*fc) with respect to /, then multiply
through by /.)

(c)    Deduce from (b) and Weierstrass' theorem that, for every continuous function
/e ^(1), the sequence (Bn,/) converges uniformly to /on I.

10.   Let X be a metrizable compact space, let (/,)„ť o be a sequence of complex-valued
continuous functions on X, and let (cn)n250 be a sequence of complex numbers.
(a)   In order that there should exist a complex measure jit on X such that ju(/rt) = cn
for all Ť, it is necessary and sufficient that there should exist a number A > 0 with the
following property: for every finite sequence (^k)o^k^n of complex numbers, we have

(Use the Hahn-Banach theorem.)

(b)   Suppose that the fk are real-valued, the numbers ck real, and/0 = 1. Then there

exists a positive measure ju, on X such that fj.(fn) ~ cn for all n if and only if, for each

n

sequence (Xk)0^kŁn of real numbers such that J]Xkfk(x) ^0 for all jceX, we have

*=o

V A* cft > 0 (cf. Section 13.3, Problem 2).

fc = 0

11.   In Problem 10, take X= [0, 1] and /„(/) = /" ("Hausdorff's moment problem").
For each sequence (cn) of scalars, put

(a)   Show that there exists a complex measure p, on X such that //.(/,) = cn for all n
if and only if there exists a number A > 0 such that

it)

for all n.

(Observe that A*cn must be the value of p for the polynomial /"(I — t)k, and use
Problem 9(b) by remarking that, for each real polynomial P, there exists a constant
CP such that

Bn,P~-Cp^P^Bn,pH-icP
n                          n

for all n.)

(b)   There exists a positive measure p, on X such that p,(fn) = cn for all n if and only

if A*cw ^ 0 for all k ^ 0 and all n ^ 0 (same method).Bohl's theorem")- (Use